The number of calls received by a car towing service average

The number of calls received by a car towing service averages 19.2 per day (per 24-hour period). After finding the mean number of calls per hour, find the probability that in a randomly selected hour the number of calls is a)exactly 2 b) more than 2 c)less then 2

Solution

Per hour is a mean of 19.2/24 = 0.8 successes.

a)

Note that the probability of x successes out of n trials is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    0.8      
          
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.143785269 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    0.8      
          
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   2   ) =    0.952577404
          
Thus, the probability of at least   3   successes is  
          
P(more than   2   ) =    0.047422596 [ANSWER]

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c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    0.8      
          
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.808792135
          
Which is also          
          
P(fewer than   2   ) =    0.808792135 [ANSWER]